Props in network theory Baez, John C; Coya, Brandon; Rebro, Franciscus
Theory and applications of categories,
01/2018, Volume:
33, Issue:
25
Journal Article
Peer reviewed
Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and ...chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with $m$ inputs and $n$ outputs is a morphism from $m$ to $n$, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the `behavior' of these circuits using morphisms between props. In particular, we give a new construction of the black-boxing functor of Fong and the first author; unlike the original construction, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Technically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.
We give an algorithm solving combined word problems (over non-necessarily disjoint signatures) based on rewriting of equivalence classes of terms. The canonical rewriting system we introduce consists ...of few transparent rules and is obtained by applying Knuth–Bendix completion procedure to presentations of pushouts among categories with products. It applies to pairs of theories which are both constructible over their common reduct (on which we do not make any special assumption).
Generic Algebras Isbell, John
Transactions of the American Mathematical Society,
1983, Volume:
275, Issue:
2
Journal Article
Peer reviewed
Open access
The familiar (merely) generic algebras in a variety $\mathscr{V}$ are those are those which separate all the different operations of $\mathscr{V}$, or or equivalently lie in no proper Birkhoff ...subcategory. Stronger notions are considered, the strongest being canonicalness of a (small) subcategory $\mathscr{A}$ of $\mathscr{V}$, defined: the structure functor takes inclusion $\mathscr{A} \subset $\mathscr{V}$ to an isomorphism of varietal theories. Intermediate are dominance and exemplariness: lying in no proper varietal subcategory, respectively full subcategory. It is shown that, modulo measurable cardinals, every finitary variety has a canonical set (subcategory) of one or two algebras, the possible second one being the empty algebra. Without reservation, every variety with rank has a dominant set of one or two algebras (the second as before). Finally, in modules over a ring $R$, the generic module $R$ is shown to be (a) dominant if exemplary, and (b) dominant if $R$ is countable or right artinian. However, power series rings $R$ and some others are not dominant $R$-modules.
PDF
